Nonconcave Portfolio Choice under Smooth Ambiguity

Abstract

We study continuous-time portfolio choice with nonlinear payoffs under smooth ambiguity and Bayesian learning. We develop a general framework for dynamic, non-concave asset allocation that accommodates nonlinear payoffs, broad utility classes, and flexible ambiguity attitudes. Dynamic consistency is obtained by a robust representation that recasts the ambiguity-averse problem as ambiguity-neutral with distorted priors. This structure delivers explicit trading rules by combining nonlinear filtering with the martingale approach and nests standard concave and linear-payoff benchmarks. As a leading application, delegated management with convex incentives illustrates that ambiguity aversion shifts beliefs toward adverse states, limits the range of states that would otherwise trigger more aggressive risk taking, and reduces volatility through lower risky exposure.

Figures (8)

• Figure 1: Manager's payoff function with $\alpha=0.5, \delta=0.2, K=1, C=0.02$.
• Figure 2: Manager's original and concavified objective functions with $\alpha=0.5, \delta=0.2, K=1, C=0.02$.
• Figure 3: Optimal terminal wealth $W_T^*$ as a function of $\xi_T$ for three relative risk aversion ($\mathrm{RRA=1-\alpha}$) levels (initial wealth $w=10$).
• Figure 4: Optimal risky proportion vs. wealth $W^*_t$ for three relative risk aversion ($\mathrm{RRA=1-\alpha}$) levels (initial wealth $w=10$, $t=T-1$).
• Figure 5: Optimal terminal wealth $W_T^*$ as a function of $\xi_T$ for three ambiguity aversion levels ($\mathrm{RAA=1-\lambda}$) in power-power case (initial wealth $w=10$, $\mathrm{RRA = 0.5}$).

Theorems & Definitions (16)

• Definition 3.1: Utility function
• Definition 3.2: Concave envelope
• Proposition 4.1: Robust representation form of smooth ambiguity preferences
• Theorem 4.1
• Example 5.1
• Theorem 5.1
• Proposition 5.1: Power--power specification
• Proposition 5.2: Exponential--power specification
• Lemma A.1
• proof